Let \(\Omega\) be the unit square \([0,1]^2\). Consider the following problem defined over \(\Omega\):
\[ \begin{cases} \frac{\partial f}{\partial t}-\Delta f = u &in \ \Omega \times [0,T] \\ \quad f = f_0 &in \ \Omega, \ t=0 \\ \quad f = 0 &on \ \partial \ \Omega,\ t>0 \end{cases} \]
where \(f_0 = \sin( 2\pi x)\sin(2\pi y)\) is the initial condition, \(u(x,y) = 8\pi^2 \sin( 2\pi x)\sin(2\pi y) e^{-t}\) is the forcing term and \(\partial \Omega\) is the boundary of \(\Omega\) where we have prescribed homogeneous Dirichelet boundary condition for all \(t\) grater than \(0\). The exact solution of the previous problem is \(u_{ex}(x,y) = \sin(2\pi x) \sin(2 \pi y) e^{-t}\) whose trend is shown in the following video.